Communications Beyond RC Limit

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

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Spoof Plasmon Interconnects—Communications Beyond RC Limit Soumitra Roy Joy , Mikhail Erementchouk, Hao Yu, Senior Member, IEEE, and Pinaki Mazumder , Fellow, IEEE

Abstract— The inception of spoof surface plasmon polariton (SSPP) mode realized in planar, patterned conductors to manage light beyond diffraction limit at a chosen frequency garnered significant attention of late. We show that, an SSPP channel can be chosen to act in two distinct ways: first, as a regular RC limited electrical interconnect at low frequencies; and second, as an exotic, beyond RC limit communication channel near its resonant frequency by binding the electromagnetic field on its surface to the elimination of capacitance C. A dynamic transformation between these two modes can constitute an energy economic, tera-scale inter-chip hybrid communication network. We have investigated theoretical limits on the information transfer capability of SSPP interconnects. We show that, a geometry dependent tradeoff relation between cross-talk limited bandwidth density and information traveling length emerges in SSPP-based communication networks. According to our analysis, a bandwidth density of 1 Gbps/µm is attainable in SSPP communication network with ∼10-mm information transfer distance, where each channel can carry ∼300-Gb/s information with nominal crosstalk. Index Terms— Interconnect, spoof plasmon, terahertz, information capacity, bandwidth, thermal noise.

I. I NTRODUCTION

T

HE world is being driven by an ever increasing demand for the distribution of data-intensive content at an enormous speed of information transfer. Large data-centers and high-speed broadband network came into being in order to cope with this vive of information explosion. For example, the bandwidth of chip-to-chip interconnects between LSIs (large-scale integrated circuits) will double every two years and is expected to reach 10 Tbit/s by 2020 [1]. It is also surmised that by the late 2020s, conventional electrical wiring will not be able to provide any practical solutions [2].

Manuscript received March 8, 2018; revised July 27, 2018; accepted September 23, 2018. Date of publication October 8, 2018; date of current version January 15, 2019. This work was supported in part by the Air Force Office of Scientific Research under Grant No. FA9550-16-1-0363 and in part by the National Science Foundation under Grant No. ECCS 1227879. The associate editor coordinating the review of this paper and approving it for publication was W. Shieh. (Corresponding author: Soumitra Roy Joy.) S. R. Joy, M. Erementchouk, and P. Mazumder are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). H. Yu was with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798. He is now with the Electrical and Electronic Engineering Department, Southern University of Science and Technology, Shenzhen 518055, China Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCOMM.2018.2874242

The trouble of conventional electrical interconnect is that their bandwidth is limited by 1/RC, where R is the resistance and C is the combined capacitance of the data-channel and that of the chip’s pin being driven. As the distance L is scaled up by a factor of α, the bandwidth of the channel 1/RC scales down by a factor of 1/α2 . As explicated by Meindl, the aggressive scaling of devices leads to a Beyond Moore’s Law era, marked by ‘tyranny of interconnects’ [3], where interconnects supplant logic devices in the role of the main factor determining circuit performance. It is predicted that the decisive role of interconnects in circuit characteristics would only strengthen with every future technology node. There also exists an alternative interconnect implementation based on optical communication link [4], renowned for its gigantic bandwidth of tens of Tbps [5], [6]. However, conventional photonic technologies are quite different from those of electronics in terms of materials, device sizes, and fabrication processes and thus lacks high density photonic device integration [7]–[10]. Because of the large thermo-optic effect in silicon compared to the electro-optic effect, the projected advantage of power economization in chip-level optical interconnect almost nullifies once we account for the power consumed by the thermal stabilization circuit [11]. In addition, the high overhead energy cost associated with electro-optic signal conversion, large (hundreds of wavelength) foot-print of optical modulator [12] and weak coupling of far-field radiation with waveguide for traversing distances smaller than 10 cm render optical interconnect approach almost prohibitive [13], [14]. According to Cho et al. [15], the critical length below which optical interconnect is energetically less-favorable than its electrical counterpart is 15 cm for a typical coupling loss of 3 dB at 6 Gbps data rate even with the assumption of an ideal modulator. Thus, we are left with a range of distance of 0.1−10cm, dubbed as the ‘last centimeter barrier’, which falls into the category of inter-chip distance and looms insurmountable with any available cost-effective means [16]. Communications through electrical interconnects are simple as they carry baseband signals, and therefore they are preferred when energy cost of data transfer is of more concern than the bandwidth of communication, whereas optical interconnects are superior when speed of communications takes over all other factors. What if we have communicating agents that prioritize different factors (i.e., energy and bandwidth) at different times? Is it possible to make an interconnect system

U.S. Government work not protected by U.S. copyright.

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that can be chosen to make act as quasi-electrical at a time, and as quasi-optical at some other time? With this question in mind, we investigate the potential of spoof surface plasmon polariton (SSPP) interconnects. With feature sizes of few tens of microns, which coincide with the dimensions of the wire pitches on electrical backplane limited by the geometric size of the chip pins and pads, we consider this alternative technology to be a new paradigm in inter-chip communications. Since the analysis of electrical means of information transfer via conductive SSPP interconnects provided with a metallic ground are pretty straightforward and known, we aim to discuss mainly on how to harness the potential of strong field confinement in the quasi-optical mode of channel to design a beyond-RC delay limited medium with optimized channel density design guideline. Coined by the seminal work of Pendry and coworkers [17], [18], spoof surface plasmon polariton (SSPP) is a special optical mode where the artificial plasma frequency are design specific rather than material specific. In addition, the modal volume of SSPP can be squeezed far below the Abbe diffraction limit without requiring the electromagnetic field penetrate into the metal. Thus we can, via SSPP, guide a signal through a sub-diffraction space while Ohmic loss can be minimized substantially. Despite deemed as a promising solution with few possible morphological variations [19]–[26], the very large cross-sectional size of the metamaterial as originally proposed posed a tremendous challenge for fabrication of SSPP structures for engineering applications, particularly for data processing and routing. Shen et al. [27] and Shen and Jun Cui [28] showed that, the very idea of SSPP mode is supported even by an ultra-thin film structure, with a thickness 600 times smaller than its operating wavelength. This discovery immediately broadened the scope of building truly sub-wavelength structure which can be bended and twisted to lend the signal routing through them in a flexible way [29]. Later on, one of the authors of this paper, Hao Yu and his group demonstrated sub-wavelength size standalone interconnect operating at THz range implemented in thin, planar CMOS compatible SSPP metamaterial [30]–[32]. This revealed the potential of SSPP interconnect to become a novel way of data transfer media between chips at THz speed and beyond. Because of the nature of field confinement around an SSPP channel, the channels in a network can be designed to remain essentially isolated from each other in a compact design scheme. In fact, the bandwidth limitation on SSPP channel derives from a quite different physics, it is the resonant frequency of the unit cell (which in turn depends on the geometry of the cell) of the metamaterial that defines the bandwidth. Therefore, within a characteristic length scale, the bandwidth of SSPP channels, unlike that of conventional electrical interconnects, can be invariant with the distance. Thus, given the fact that the exclusive research on terahertz technologies increased DC-THz conversion efficiency of silicon integrated THz sources almost 1000 times since 2008 to date [33], [34], spoof plasmon interconnects can be a very promising choice for the future tera-scale data transfer between chips, amassing the good sides of both electrical and optical interconnects. On the one hand, the SSPP interconnect possesses the great advantage of CMOS compatibility and

its quasi-electrical mode can transfer data without a modulation scheme and thus liberate from overhead energy cost of source driver, modulator’ and detectors’ power. On the other hand, like optical ones, it may accommodate beyond RC limit communication where bandwidth per channel can be few hundreds of Gbps for few tens of mm information propagation length, in addition to the possibility of ultra-low energy (subfemtojoule) [35] signal detection through facilitating ‘quantum impedance conversion’ [36] of photon. We may switch between these two different types of information transfer mode in SSPP interconnect, namely electro-SSPP and opto-SSPP mode, which may offer an energy-economic data transfer solution between chips where the data traffic volume may vary by orders of magnitude at various period of time. We showed that, with our prescribed geometry, an SSPP based data-bus can provide ∼ 1 Gbps/μm bandwidth density while the signal can propagate upto state-of-the-art chip edge length ∼ 10 mm, where each channel would carry information at approximately 300 Gbps rate without invoking significant cross-talk. The present work demonstrates the fundamental interplay between various figure-of-merits of information capacity of an inter-chip communication network comprising SSPP channels. In section II, we discuss the bandwidth density maximization strategy in an SSPP network. In section III, we define the characteristic length of information propagation pertaining to SSPP channel and its relation to the geometry of the channel. Section IV shows the relation between finite information propagation length and the resistivity of SSPP channel, which eventually led to the estimation of noise property and Shannon information capacity in the limit of thermal noise contributed by the SSPP channels. Section V discusses the quantitative comparison between various alternatives of chip-to-chip interconnects in terms of cross-talk in the system. II. BANDWIDTH IN C ROSS -TALK M EDIATED SSPP C HANNELS Figure 1a is a schematic of the data transfer system that we propose for inter-chip communication where the data-bus consists of SSPP channels running in parallel. A pair of channels are formed by placing two SSPP waveguides back to back. Each of such channel-pair maintains a guard space between the nearest face-to-face SSPP waveguides to avoid cross-talk. An SSPP mode possesses a continuous band, starting from zero frequency- where there is no field confinement and hence the channel is subjected to significant cross-talk with the neighboring channels; upto the spoof plasma frequency- where the degree of field confinement is maximum and the channels can be decoupled from each other (Fig. 1b). Between these two extremities, the field localization property undergoes gradual changes. Figure 1c shows the dispersion diagram of an isolated SSPP channel. In order to generate a dispersion diagram independent of geometric dimensions of interconnects, we, in Fig. 1c, instead of plotting the absolute value of frequency ω and propagation constant β, choose to plot a unit-less frequency parameter ω ˆ = 2ωh/πc against another unit-less propagation parameter βˆ = βd/π. At a given frequency ω,

JOY et al.: SPOOF PLASMON INTERCONNECTS—COMMUNICATIONS BEYOND RC LIMIT

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high resolution imaging. The other region is the soft SSPP regime, where the field localization is moderate, however the bandwidth of this region is appreciably large. As soft SSPP regime can offer both large bandwidth and field confinement together, this can be leveraged towards making interconnect channel with high information capacity which may defy RC limited bandwidth. Let us begin our analysis of an SSPP interconnect system by designating the groove length, width, period, thickness and channel separation as h, a, d, 2b and 2t, respectively. It is shown in other works that the thickness (2b) of SSPP waveguide plays a trivial role in determining dispersion relation. This helps to divide the SSPP system in two independent 2D sub-systems for the sake of simpler analysis. The first subsystem comprises of a periodic groove of length h and period d on zx-plane where the subsystem is infinitely extended along y-direction. The second subsystem consists of a periodic array of metal slabs of small thickness 2b, separation a, and period d on xy-plane where the subsystem is considered infinite along z-direction. Taking the midpoint of the groove as the origin of co-ordinates, he field distribution on the surface of the first sub-system can be written as follows. (d) (z) = Aan e−κc,an (z+t) Eax n

(d) Ebx (z)

=

Abn eκc,bn (z−t)

n

iβan κan Aan e−κc,an (z+t) 2 − β2 k an n iβbn κbn (d) Ebz (y) = Abn eκc,bn (z−t) 2 − β2 k bn n (d) Eaz (z)

Fig. 1. (a) An inter-chip communication network employing SSPP channels. Scales of interconnects are shown in terms of free-space wavelength (λ) corresponding to spoof plasmon resonance, a typical wavelength being 300 μm. (b) The left figure is the top-view of electric field distribution in SSPP channels, while the right figure is a cross-section view of the field, showing modal confinement. The simulation frequency is ∼1 THz, roughly the spoof plasmon resonance frequency of an SSPP interconnect with groove length 80 μm. (c) Comparison between dispersion relations of thin SSPP structures for two different cases: channel suspended in air, and that fabricated on a silicon substrate. The corresponding basic SSPP structure is shown in the lower right corner. The solid lines are from the developed theoretical model, while the discrete dots (•) and (◦) are obtained by FDTD numerical simulation in COMSOL Multiphysics [37].

any change in the geometric dimension of SSPP interconnect would cause a corresponding change in the absolute value of β in such a way that the relation between the normalized parameter ω ˆ and βˆ will remain unaltered, as they absorb the effect of geometric change into themselves. We may distinguish two different regions in the dispersion curve of SSPP channels. One of them is the strong SSPP regime, where the dispersion curve becomes almost flat and provides deep sub-wavelength field confinement, which can be leveraged for

=−

(1)

where Aan and Abn are the complex amplitude of n-th mode of face-to-face posing SSPP waveguides ‘a’ and ‘b’, respectively, β is the propagation constant, and κc is the attenuation constant of evanescent field along z direction. κc can provide a measure of modal confinement that we shall capitalize to suppress cross-talk, and we shall be discussing about it in detail shortly. The magnetic field components outside the groove can be expressed as iω0 κan (d) Hay (z) = − Aan e−κc,an (z+t) 2 k 2 − βan n iω0 κbn (d) Hby (z) = − Abn eκc,bn (z−t) (2) 2 − β2 k bn n Each of the above components has eiβx dependence along the propagation direction x. The dispersion relation of the sub-system is [38], a aβ0 ) tan(Qh) (3) β 2 − Q2 = Q sinc2 ( d 2 where Q is the wave-vector inside the grooves along their length, and for infinitely extended subsystem (2b → ∞) along y direction, Q = k0 = ωc . The relation of Q ≈ k0 holds even at the limit of vanishing thickness b of SSPP interconnects. Therefore, analysis of surface mode for the second subsystem with finite thickness can be postponed until we begin section IV that discusses thermal noise in SSPP interconnects.

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The length of modal confinement (tc ) of SSPP modes can be 1 1 = 2 , where ω is angular estimated as tc = 2κc 2 β − ( ωc )2 frequency, and c is the speed of light. In order to transfer data with suppressed inter-channel coupling (less than −3 dB), we may choose to utilize the upper fc fraction of the bandwidth (BWsp ) of an SSPP channel where modal confinement a ωmin ωmin ng,ef f tan( ng,ef f h), is stronger. Then κc,min ≈ d c c where ωmin = 2π(1 − fc )BWsp , and ng,ef f is the effective refractive index of the medium adjacent to the channel. Hence the minimum amount of space required to interleave between a pair of neighboring SSPP channel is

determined by considering

tc,max /h =

2 c d 1 2 . 2a 1 − fc 2πng,ef f h.BWsp

(4)

The bandwidth BWsp is itself a function of the groove geometry, particularly governed by groove length, as [38]: π c − δw ) 2 2πng,ef f h π a d c , − √ = 2πng,ef f h 2 d 4h2 − d2

BWsp = (

(5)

An important figure of merit for an interconnect network is the bandwidth density (ρBW ), defined as the amount of bandwidth available for data transfer per unit cross-sectional length of the data-bus at reduced cross-channel coupling. It has been experimentally demonstrated that, by placing two SSPP waveguide back-to-back with nominal separation, the cross-talk between them can be kept below −20 dB [31]. So, the critical separation which would determine the channel packing density is the separation between two SSPP waveguide placed face-to-face. Bandwidth density of SSPP data bus of a design shown in Fig. 1a can be expressed as cfc 2fc .BWsp ≈ tc,max + 2h 2ng,ef f h(tc,max + 2h) c fc (1 − fc )2 = , 2 4ng,ef f h (1 − fc )2 + D

ρBW =

d

(6)

where D = 2 . As evident from Eq. 6, choosing a π a substrate with low refractive index (ng,ef f lower) is beneficial for the design of a closely packed SSPP network for a given bandwidth of data transfer. In addition, it shows that reducing groove length h leads to a greater bandwidth density. The lowest limit on h is dictated by the condition on the ratio of geometric length of the structure to support spoof plasmon mode, which is 2h > d [38]. The dependence of ρBW over fc (i.e., fraction of total SSPP bandwidth BWsp which is cross-talk suppressive) is also worth looking at. For small fc , cross-talk limited bandwidth linearly increases with fc , however, ρBW quadratically decreases as fc approaches its maximum value of 1. Its optimum value fc,opt can be

∂ρBW = 0 that yields: ∂fc √ (D + D2 + D3 )2/3 − D √ fc,opt = 1 − (D + D2 + D3 )1/3 ⎧ 2 3 ⎪ ⎨ − δ, when D = 1 + δ s ≈ 15 50 2 ⎪ ⎩ (1 + √ ), when D >> 1 3 3 D

(7)

In this limit of large D, an interesting relation of (fc ≥ 13 ) emerges, which indicates that we can have at least 33% of the total bandwidth of the SSPP channel available for cross-talk suppressive data transfer. If we plug in the nominal value of fc = 13 in Eq. 6, then the maximum attainable bandwidth density in SSPP network is found to be ρBW |max ≈ 1 π2 c . ng,ef f h2 12π 2 + 27d/a Figure 2a gives the visual depiction of bandwidth optimization strategy. It shows that, we may utilize the upper 33% of each of the SSPP channel bandwidth to maximize the bandwidth density (ρBW ). As the condition of existence of SSPP warrants hmin = d/2, we can increase bandwidth density ρBW further by reducing periodicity d. However, a reduction of d would increase the mode coupling factor ad , which would adversely affect the BWsp . In addition, reduction of h would also move the system into higher operating frequency, where the Ohmic loss in conductor through the skin effect would creep in and limit signal propagation length. Section 3 deals with the question of how far the information can travel through a channel consisting of a real metal with finite conductivity. One other interesting feature of SSPP channel is its nature of confinement along the vertical (y) direction, which is in fact stronger than its lateral confinement and can be shown to be t−1 c,max = 2κc,min ≈ 2βmin ≈

π(1 − fc ) ng h

(8)

where κc is the vertical evanescent wavevector along y direction. If we stack SSPP channels one on top of the other with interleaving space between them equal to the vertical confinement length (tc,max ) of the mode, then the bandwidth density ρBW of the vertically stacked channels becomes, πc ρBW ≈ fc (1 − fc ) (9) 4n2g h2 Comparison of bandwidth density for planar SSPP channel cluster and that for vertical channel cluster written in Eq. 6 and 9 respectively reveals an interesting fact. Equation 9, unlike Eq. 6 is free from the term d/a. This adds a relative flexibility in designing the groove width a without affecting the bandwidth density. In addition, Eq. 9 shows that the optimal fraction of SSPP bandwidth in cross-talk suppressive regime, in case of vertical channel stacking, can be 50%, significantly higher than that achievable in planar arrangement of channels. III. T RAVELING L ENGTH OF SSPP M ODE IN L OSSY M ETAL Following [39], we can incorporate metallic loss in our analysis of SSPP channel for metal with finite conductivity.


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where Q = Im[Q]. The regime β >> Q0 ng can be seen in an SSPP structure with groove length (h) significantly larger than half of the period (d). It should be emphasized that the structures considered in the present work are characterized by h < d, and therefore do not operate in the strong decay regime. While a knowledge of lp at a single carrier frequency ωc may suffice to estimate the propagation distance of a message signal with bandwidth Δωm as long as ωc >> Δωm , it is not quite useful in case of information transfer over SSPP channel network, where the carrier frequency is supposed to be nearly comparable to the signal bandwidth. Under such circumstances, we find it worthwhile to introduce a slightly different characteristic length lt which is defined as the length that a pulse with rectangular Fourier spectral profile of bandwidth Δωm can travel until the power distributed over its bandwidth reduces by half. This leads to following relation of power loss in metal, ωmax A2 Δωm (12) A2 e−2βI (ω)lt dω = Ploss = 2 ωmin where βI = Im[β]. If we operate in the weak SSPP regime for information transfer, it can be seen from Eq. 10 and 11 that, for ad > Q0 ng 2a d2

lt =

a c h 1a = 2 ls ωp (1 − 0.5fc ) ls π(1 − 0.5fc)

(15)

where ωmax ≈ ωp is assumed, and Δωm = fc ωp is substituted. It is obvious from the expression of lt that, for a given information bandwidth Δωm , the larger the length of the groove h of SSPP we choose, the further the signal can travel. The expression for bandwidth density and information propagation length reveals a fundamental trade-off about choosing groove length h. While choosing smaller h increases cross-talk limited bandwidth density of SSPP data-bus, it decreases signal traveling length through the channel. Figure 2b illustrates this trade-off, drawn for waveguide whose geometry is varied in a way so as to keep all relevant aspect ratios fixed at some optimal value (i.e., h/d ∼ 12 , a/d = 0.1) and the groove length h is varied. The corresponding maximum bandwidth density of SSPP data-bus and the traveling length of information at that maximum bandwidth density is plotted on the same graph. It can be seen from the diagram that, for interconnect length to be ∼ 10 mm, which is roughly the state-of-the-art chip edge length, we can obtain ∼ 1 Gbps/μm bandwidth density with an optimal geometry design of SSPP waveguide. Information carried by each channel can be determined as

604


bandwidth density multiplied by an average space occupied by an individual channel including the guard space, which is found to be 300 Gbps. In order to determine the upper bound of signal propagation limit, we choose gold while plotting the numerical value of Fig. 2b which has a better conductivity over other commonly used metals. For a different choice of metal with conductivity σm , the numerical value of information transfer length can be found

by scaling the value determined σm √1 . in Fig 2b by a factor of σAu , since skin depth ls ∼ σ 1 If the optimal value of fc is taken as as discussed previously, 3 the corresponding value of information traveling length will be 6 a lt,opt = h. 5π ls

characteristic length ltr . Thus the resistance (RSP ) of SSPP channel per unit length can be found from the following: 1 ωp Px,tot (ω)dω (18) |Ix |2 RSP lt Δωm = 2 ωp −Δωm

IV. I NFORMATION C APACITY IN THE L IMIT OF T HERMAL N OISE In the regime of low-cross-talk, the main limitation on the Shannon information capacity of a channel originates from signal to noise ratio (SNR) in the channel. In the present work, we are particularly interested in the thermal noise characteristics of SSPP channels, which is expected to be the most dominant type of noise over others (such as flicker noise and shot noise) at higher operating frequency and low power transmission. If SNR of SSPP channel is limited by thermal noise, we may want to determine the equivalent electrical resistance of SSPP channel in order to estimate thermal noise. To this goal, we calculate the power (Px,tot ) propagating in the longitudinal (x) direction on a planar SSPP channel on each of its side and equate the fraction of the power lost in joule heating process with I 2 R loss. Because of the boundary condition on the plane of SSPP channel made of metal of good conductivity, it can be argued that the in-plane component Ez is negligible compared to the other component Ey of electric field. Hence power spectral density Px,tot is mainly contributed by Ey and Hz and can be found as follows. Hz,a = Aeiβx e−κ(y−b) eiQz β Aeiβx e−κ(y−b) eiQz Ey,a = ω ∛ 1 h ∗ Px,tot (ω) = Re[Ey,a Hz,a ]dy 2 z=0 y=b 1 β 2 dz = A h (16) 4 ωκ The set of relations in Eq 16 describes the second subsystem, discussed in section II, comprised of metal slabs of infinite length and 2b thickness that accounts for field distribution in the region defined by y ≥ b. Here A is the √ amplitude of magnetic field with the SI unit of Amp / meter / Hz, and = n2g,ef f is the effective homogeneous permittivity of the medium mediated by SSPP channel. The surface current in the longitudinal direction is h Jx dz ≈ hHz,a (x)|y=b = Aheiβx (17) Ix (ω) = 0

If a pulse of bandwidth Δωm is to be transferred through the SSPP channel, half of the power will be lost after traveling the

where we used the spoof-plasma frequency ωp in the limit of integral as an approximation of BWsp . If we denote the wave-vector along the groove length as Q, then using the fact that Q ≈ ω/c in the relation β 2 − κ2 + Q2 = (ω/c)2 describing the dispersion characteristics of the field over the SSPP surface, we can set β ≈ κ. Thus we obtain, 1 1 ωp ln Rsp = 8lt hΔωm ωp − Δωm 1 1 = | ln(1 − fc )| (19) 8lt hΔωm where Δωm = fc ωp is used. Then the electronic noise generated by the thermal agitation of electrons on patterned metal of the channel can be modeled as a noiseless resistor in series connection with a noise voltage source v¯n as follows. v¯n2 = 4kB T Rsp lch Δωm /2π

(20)

where lch = N d is the channel length, N being the number of unit cell in the SSPP channel with the period d. In addition, kB is the Boltzmann constant and T is the absolute temperature of the channel. Now if the SSPP mode excites an rms voltage value of Vr at the signal receiving end, the Shannon information capacity (Cinf o ) of SSPP channel in the limit of thermal noise can be written as Cinf o = ρBW log2 (1 + SN R) 2πVr2 (Δωm ) = ρBW log2 1 + 4kB T Rsp lch Δωm

(21)

Vr2 is the signal to noise ratio, ρBW v¯n2 is the SSPP bandwidth density limited by cross-talk among neighboring channels and follows Eq. 6. Voltage drop Vr across the receiver resistance Rr is related to the transmitted information bandwidth Δωm as follows. where SN R =

Vr2 = Pr Rr Δωm 1 lch 1 1 2 A h ln(1 − )(1 − ) Pr = 4 ωp 2 lt

(22) (23)

where Pr is the power at the receiving end. Characterization of SSPP channel resistance Rsp facilitates determining the bit error rate (BER) due to channel induced noise. Out of two types of noise, namely, thermal noise and shot noise, let us assume that thermal noise prevails over shot noise (i.e., thermal noise power is at least twice of shot noise power), and the receiver can be controlled in a way so as not to contribute to the thermal noise. In addition, let us also choose to design receiver input resistance (Rr ) to be an order of magnitude larger than the channel resistance 1 (Rch = Rsp N d = Rr ), so that we may have large voltage r drop (Vr ) at the receiving end. Shot noise power can be written Vr . Thermal noise power can be as 2qIr Δf Rr , where Ir = R r


1 . Then the above stated condition Rr leaves us with the following inequality condition

written as 4kB T Rch Δf

4kB T RchΔf Vr ≥ 2 × 2q Δf Rr Rr Rr

(24)

These conditions set an upper limit for the signal voltage kB T (Vr ) at the receiving end as Vr ≤ , where r is the ratio rq of resistance of the receiver to the channel, q is the charge of electron. Since SSPP channel is not suitable to carry DC current, as at ω = 0 there would be no field confinement, we must choose a source encoding scheme that is devoid of any DC offset. Let us choose non-return-to-zero (NRZ) coding for digital signal transmission where bit-0 and bit-1 are represented by negative and positive pulses of equal amplitude. Since we also want to maximize the information capacity by kB T to denote choosing optimal SNR, let us set ±Vr,max = ± rq the mean value for binary bits, and zero as the threshold value. Since NRZ encoding is symmetric for 0 and 1 bit, the probabilistic bit error rate can be calculated as ∛ (V + V )2 1 r dV exp − BER = √ 2vn2 vn 2π 0 V 1 r √ , (25) = erfc 2 vn 2 where Vr = Vr,max is chosen for maximum SNR. Then for the utilization of fc fraction of SSPP bandwidth for data transmission with suppressed inter-channel coupling, the BER can be related with the geometry of the SSPP waveguide as follows. 1 lt √ πkB T 1 h (26) BERsp = erfc 2 rq | ln(1 − fc )| lch If fc = 13 is chosen for the reason of bandwidth density maximization as described in previous section, and channel length is taken as lch = lt , and r = 10 is considered, then the optimal BER turns out to be BERsp,opt = 1 1 πKB T √ erfc h . At the same optimal condition 2 5q 2 ln(3/2) for data transfer within thermal noise limit, the optimal Shannon information capacity can be written as Cinf o,opt = ρBW,max log2 (1 + SN Rmax ) π2 c 8πhKB T 1 = log2 1 + 2 d ng,ef f h 25q 2 ln(3/2) 12π 2 + 27 a (27) where SN Rmax is the maximum SNR possible to obtain at the receiver while still operating in the limit of thermal noise generated by SSPP channel. Figure 3a shows how the Shannon information capacity and BER of an SSPP channel at room temperature vary with the groove length while SNR is maximum at the receiving end within the limit of thermal noise generated by SSPP channel.

605

V. SSPP I NTERCONNECT IN C OMPARISON W ITH OTHERS : T HE B ENEFIT OF M INIMIZED I NTERFERENCE In this section, we will make a comparative discussion to facilitate choosing the means of interconnects depending on system requirement. We will make an analytical estimation the interference suppression in a parallel bus system made of conventional electrical interconnects, that of optical interconnects, and that of our proposed SSPP interconnects.1 A. Cross-Talk in Optical Interconnect A simple assumption of information transfer in the form of TE mode propagation in a dielectric slab may suffice for cross-talk estimation in a system of optical interconnects. The dispersion relation for the even TE mode in an isolated dielectric slab is κ = kz tan(kz w)

(28) 2 2 2 where 2w is the thickness of the slab, kz = k0 nd − β 2 2 2 and κ = β − k0 na are the transverse wave-vector inside and outside of the dielectric waveguide, respectively, and nd and na are refractive index of the slab and the environment, respectively. Coupling between identical neighbor waveguide pair, with the help of Eq. 28 is determined as follows [42], [43], 3w+2t ω 2 2 e−κ(z−w) Kop = 0 (nd − na )Ea0 Ed0 4 w+2t × cos(kz (z − 2w − 2t)) dz ω E2 2 = 0 d0 cos (k w) κ(1 − e−2κw ) z 4 k02 + (1 + e−2κw )kz tan(kz w) e−2κt =

2 ω0 Ed0 cos2 (kz w)κe−2κt 2 k02

(29)

where 2t is the separation between neighboring interconnects, √ k0 = ω 0 μ is free space wave-vector, and Ed0 (Ea0 ) is the electric field magnitude inside (outside) the dielectric slab. In the RHS of the last relation, the continuity condition of tangential electric field at the interface (Ed0 cos(kz w) = Ea0 ) is applied. We can obtain the behavior of coupling parameter in the limit of high frequency as following lim Kop =

k0 w→∛

lim

Kop 1 πc 2 −2ωnd t/c = ωlim e w c →∛ w 2ωnd w 2Kz w→π

(30)

The coupling constant Kop would enter as the non-diagonal terms of the operator Hop , where Hop describes the evolution of the modes along propagation direction in a system of optical interconnects, perturbed by the presence of neighbor waveguides. The eigenfunction analysis of Hop leads to the 1 We have not included plasmonic interconnects in our comparative discussion, as it is shown by others [41] that plasmonic interconnects are meant for communication within 10 μm distance, whereas we are concerned about centimeter scale distance coverage, a typical separation between electrical chips on board

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following relation of a1 (x), where a1 (x) is the mode amplitude of a waveguide due to unity amplitude signal injection into either one of its nearest neighbor interconnects. √ 1 −i a1 (x) = √ (eixβ− − eixβ+ ) = √ eiβ0 x sin(Kop 2x) 2 2 2 (31)

In order to estimate cross-talk, we shall, as we did previously, inject a unit signal at the input terminal of one particular interconnect, and observe how the signal victimizes the neighboring channel. However, since electrical system incorporates the parameter R (resistance per unit length) in the coupling term, which also leads to signal absorption, we shall define the 3 dB cross-talk length lx,el in a way that satisfies the following relation.

We can define cross-talk length lx as the length at which 1 |a1 (lx )| = √ . This definition leads to the following expres2 sion of lx . π 1 lx,op = √ 2 2 Kop

(32)

Combining Eq. 30 and Eq. 32, we get the following expression of critical cross-talk length in optics at high frequency regime. √ w 2 ωnd w 2ωnd t/c lim lx,op = ωlim e (33) wω w c →∛ π c c →∛ The above relation is expected to be reasonably correct for πc 1. The expression of Ksp can be further simplified if we are ω interested at the frequency regime, where 2 c → π (i.e., h ω π c = − δ, where δ is a small number). In that limiting h 2 case, following from the dispersion relation in Eq. 3, we obtain π a1 1d κ = limδ→0 , and β = limδ→0 κ(1 + δ) ≈ κ. 2h d δ 2a |Ksp | = Then the coupling parameter Ksp becomes lim 2ω c h→π 2tn2ef f κ2 exp(−2κt). The eigenvalues of the matrix β are β± = β0 ± Ksp . It can be shown that, owing to unit signal injection at SSPP channel ‘a’, its neighbor ‘b’ will register a −3 dB interference at a length of lx,sp which abides by the following relation, 1 iβ l iK l 1 −iKsp lx 0 x sp x (42) (e −e ) = √ |b(lx,sp )| = e 2 2 which eventually results in the following, lx,sp =

π 1 4 |Ksp |

(43)

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2ω c → π, critical cross-talk length In the limiting case of h in SSPP interconnects become h2 d 2 π ωh 2 lim lx,sp = lim − 2ωh→πc 2ωh→πc 2πtn2 2 c ef f a π a 2tc × exp (44) h d πc − 2ωh The above expression of critical cross-talk length in SSPP interconnects system reveals that, lx,sp diverges at a frequency πc . Thus we can achieve cross-talk suppression at some ω = 2h chosen high frequency by tailoring the geometry of SSPP structure, particularly the length h of the groove. Comparison of Eq. 33 and Eq. 44 reveals that, SSPP interconnect can mimic the behavior of optical interconnect upto some design specific frequency in the sense that, the cross-talk suppression in both of the types of interconnects gets better with the choice of carrier of higher frequency. Figure 3b illustrates the critical length in different interconnect systems beyond which cross-talk becomes significant. A typical geometry of SSPP interconnects system is chosen with groove length of 20 μm and period of d = 20 μm. The spacing between neighboring interconnect is taken as t = 20 μm. For a fair comparison, an electrical interconnect system and a SiO2 core based optical interconnect system of similar geometric width and interconnect spacing are also considered to determine their respective cross-talk limited length. It shows that electrical and optical interconnects perform better at low-end frequencies and high-end frequencies, respectively. SSPP interconnect can demonstrate semi-optic behavior of divergent cross-talk length at frequencies close to spoof plasmon resonance. We figure out two different kinds of mode that can be supported by SSPP interconnect, namely electro-SSPP and opto-SSPP mode. The electro-SSPP mode of spoof plasmon interconnects has remained largely unexploited. In order to appreciate the electrical behavior of SSPP interconnects, consider Fig. 3c where we illustrate that SSPP interconnect can support both electrical and THz-optical mode of communication. Provided with a metallic ground plane, we can utilize SSPP interconnect in a way similar to pure electrical means of communication, where the signal is sent in the form of transverse electromagnetic (TEM) mode confined between the metallic wires and ground. The merit of a patterned metal wire (i.e., SSPP interconnect) in terms of bandwidth and cross-talk suppression doesn’t change appreciably compared to that of an un-patterned, plain wire since the product of resistance and (xd being the dielectric capacitance per unit length RC = σxd spacing between interconnect and the ground) is independent of the width of the wire. In addition to that, close to the resonant frequency (which can be designed to be at few THz), SSPP interconnects can transfer information in the form of surface confined spoof plasmon mode where metallic ground plane is redundant. As we showed, electrical and optical means of communication have a stark contrast of cross-talk behavior; electrical means are suitable for information transfer at low frequencies, while the optical ones are good for high frequency spectrum.

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where Eov = Esource +Emodualate +Edetect +Ecouple is the overhead energy associated with the carrier source (i.e., laser driver), modulator, detector and the coupling loss, Np is the number of bit transferred within time Δt, and Ep is the energy associated with each bit. Clearly, an optical means of inter-chip communication will be energy-economic at a time when there is a heavy traffic of information such that Np Ep >> Eov . Recall that we are concerned about centimeter scale distance coverage between electrical chips, where it is not unlikely that the volume of information transfer between them will vary in a wide range from time to time. We believe, in a system with widely varying demand of data transfer rate, SSPP interconnects can be a potential alternative of energy-economic means of communication, as they allow us to switch between two different modes of communication, namely electro-SSPP and opto-SSPP mode. VI. C ONCLUSION

Fig. 3. (a)Variation of the optimized Shannon information capacity (Copt ) and the optimized bit error rate (BER) in SSPP channel at T = 300K with the variation of groove length. The channel length is taken as the characteristic traveling length (lt ) for the corresponding groove length. The data is plotted for maximum attainable SNR at the receiver end within thermal noise limit of the channel. The geometric ratio a/d of the SSPP channel is taken as 0.1. (b) Critical length beyond which cross-talk becomes significant at different interconnect system. (c)Illustration of two different kind of modes that can be supported in SSPP interconnects. (i) SSPP interconnects, in conjunction with a metallic ground plane are operating in electrical mode. The field distribution is analogous to the TEM mode that propagates in purely electrical interconnects (shown in inset). (ii) SSPP interconnect operating in THz-optical mode. The transverse field distribution is analogous to that of optical interconnect (shown in inset).

In any case, we would prefer to minimize the energy spent for per bit of information transfer, Eb . Ep Eov +1 (45) Eb = Δt Ep Np

In summary, the present work comprises an information theory adopted for spoof plasmon channels which promise to be an energy-economic solution to traverse the broadly-discussed ‘last centimeter’ distance between two chips. This is a unique solution for centimeter scale data transfer that combines the advantages of both optical and electrical nature. The purpose of our analyses is to reveal and quantify the interplay between various parameters related to the information transfer process through a compact data-bus of spoof plasmon channel. A part of our work addresses the optimization of the bandwidth density and signal propagation length. Our analysis shows that the best way of maximizing bandwidth density is to design SSPP channels with the groove length comparable to half of the period of the structure, leading to ∼ 1 Gbps/μm bandwidth density with the signal propagation length ∼ 10 mm, and ∼ 300 Gbps speed per channel. An effective resistivity of the quasi-optic SSPP channel has been introduced and used for the analysis of the noise characteristics of the channel at a given temperature. In the limit of thermal noise generated by SSPP channel, we also quantitatively demonstrated that an interplay exists between maximization of Shannon information capacity of the channel and the probability of bit error for digital data transfer via spoof plasmon channel. We believe this work would lay a foundation for the novel and prospective technique of spoof plasmon based inter-chip communication to facilitate the next generation fast and reliable data transfer process with high signal integrity. R EFERENCES [1] B. B. Brey, The Intel Microprocessors, 8th ed. Upper Saddle River, NJ, USA: Prentice-Hall, 2008. [2] Y. Urino et al., “First demonstration of high density optical interconnects integrated with lasers, optical modulators, and photodetectors on single silicon substrate,” Opt. Express, vol. 19, no. 26, pp. B159–B165, Dec. 2011. [3] J. D. Meindl, “Beyond Moore’s Law: The interconnect era,” Comput. Sci. Eng., vol. 5, no. 1, pp. 20–24, Jan. 2003. [4] M. Haurylau et al., “On-chip optical interconnect roadmap: Challenges and critical directions,” IEEE J. Sel. Topics Quantum Electron., vol. 12, no. 6, pp. 1699–1705, Nov./Dec. 2006. [5] G. Keiser, Optical Fiber Communications. Hoboken, NJ, USA: Wiley, 2003. [Online]. Available: http://onlinelibrary.wiley.com/doi/ 10.1002/0471219282.eot158/abstract


[6] R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol., vol. 28, no. 4, pp. 662–701, Feb. 15, 2010. [Online]. Available: https://www.osapublishing.org/abstract.cfm?uri=jlt-28-4-662 [7] A. E.-J. Lim et al., “Review of silicon photonics foundry efforts,” IEEE J. Sel. Topics Quantum Electron., vol. 20, no. 4, Jul./Aug. 2014, Art. no. 8300112. [8] J. Michel and L. C. Kimerling, “Electronics and photonics: Convergence on a silicon platform,” in Proc. 33rd Eur. Conf. Ehxibit. Opt. Commun. (ECOC), Sep. 2007, pp. 1–2. [9] D. J. Shin et al., “Integration of silicon photonics into DRAM process,” in Proc. Opt. Fiber Commun. Conf. Expo. Nat. Fiber Opt. Eng. Conf. (OFC/NFOEC), Mar. 2013, pp. 1–3. [10] G. Roelkens et al., “Silicon-based photonic integration beyond the telecommunication wavelength range,” IEEE J. Sel. Topics Quantum Electron., vol. 20, no. 4, Jul./Aug. 2014, Art. no. 8201511. [11] R. K. Dokania and A. B. Apsel, “Analysis of challenges for onchip optical interconnects,” in Proc. 19th ACM Great Lakes Symp. VLSI, New York, NY, USA, 2009, pp. 275–280, doi: 10.1145/ 1531542.1531607. [12] Y. Urino, T. Horikawa, T. Nakamura, and Y. Arakawa, “High density optical interconnects integrated with lasers, optical modulators and photodetectors on a single silicon chip,” in Proc. Opt. Fiber Commun. Conf. Expo. Nat. Fiber Optic Eng. Conf. (OFC/NFOEC), Mar. 2013, pp. 1–3. [13] P. Markov, J. G. Valentine, and S. M. Weiss, “Fiber-to-chip coupler designed using an optical transformation,” Opt. Express, vol. 20, no. 13, pp. 14705–14713, Jun. 2012. [Online]. Available: https://www. osapublishing.org/abstract.cfm?uri=oe-20-13-14705 [14] G. Roelkens, P. Dumon, W. Bogaerts, D. Van Thourhout, and R. Baets, “Efficient fiber to SOI photonic wire coupler fabricated using standard CMOS technology,” in Proc. IEEE LEOS Annu. Meeting Conf., Oct. 2005, pp. 214–215. [15] H. Cho, P. Kapur, and K. C. Saraswat, “Power comparison between highspeed electrical and optical interconnects for interchip communication,” J. Lightw. Technol., vol. 22, no. 9, pp. 2021–2033, Sep. 2004. [Online]. Available: https://www.osapublishing.org/jlt/abstract.cfm?uri=jlt-22-92021 [16] Q. J. Gu, “THz interconnect: The last centimeter communication,” IEEE Commun. Mag., vol. 53, no. 4, pp. 206–215, Apr. 2015. [17] J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science, vol. 305, no. 5685, pp. 847–848, Aug. 2004. [18] F. J. Garcia-Vidal, L. Martín-Moreno, and J. B. Pendry, “Surfaces with holes in them: New plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt., vol. 7, no. 2, p. S97, 2005. [Online]. Available: http://stacks.iop.org/1464-4258/7/i=2/a=013 [19] M. A. Kats, D. Woolf, R. Blanchard, N. Yu, and F. Capasso, “Spoof plasmon analogue of metal-insulator-metal waveguides,” Opt. Express, vol. 19, no. 16, pp. 14860–14870, 2011. [Online]. Available: https://www.osapublishing.org/abstract.cfm?uri=oe-1916-14860 [20] N. Yu et al., “Designer spoof surface plasmon structures collimate terahertz laser beams,” Nature Mater., vol. 9, no. 9, pp. 730–735, Sep. 2010. [Online]. Available: http://www.nature.com/nmat/journal/v9/n9/ abs/nmat2822.html [21] D. Woolf, M. A. Kats, and F. Capasso, “Spoof surface plasmon waveguide forces,” Opt. Lett., vol. 39, no. 3, pp. 517–520, Feb. 2014. [Online]. Available: https://www.osapublishing.org/abstract.cfm?uri=ol39-3-517 [22] B. Ng et al., “Spoof plasmon surfaces: A novel platform for THz sensing,” Adv. Opt. Mater., vol. 1, no. 8, pp. 543–548, Aug. 2013. [Online]. Available: http://onlinelibrary.wiley.com/doi/ 10.1002/adom.201300146/abstract [23] S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett., vol. 97, pp. 176805-1–176805-4, Oct. 2006. [24] Z. Xu, K. Song, and P. Mazumder, “Analysis of doubly corrugated spoof surface plasmon polariton (DC-SSPP) structure with sub-wavelength transmission at THz frequencies,” IEEE Trans. THz Sci. Technol., vol. 2, no. 3, pp. 345–354, May 2012.

609

[25] Z. Xu and P. Mazumder, “Bio-sensing by Mach–Zehnder interferometer comprising doubly-corrugated spoofed surface plasmon polariton (DC-SSPP) waveguide,” IEEE Trans. THz Sci. Technol., vol. 2, no. 4, pp. 460–466, Jul. 2012. [26] S. R. Joy, M. Erementchouk, and P. Mazumder, “Spoof surface plasmon resonant tunneling mode with high quality and Purcell factors,” Phys. Rev. B, Condens. Matter, vol. 95, no. 7, p. 075435, Feb. 2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.95. 075435 [27] X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, “Conformal surface plasmons propagating on ultrathin and flexible films,” Proc. Nat. Acad. Sci. USA, vol. 110, no. 1, pp. 40–45, Jan. 2013. [Online]. Available: http://www.pnas.org/content/110/1/40 [28] X. Shen and T. J. Cui, “Planar plasmonic metamaterial on a thin film with nearly zero thickness,” Appl. Phys. Lett., vol. 102, no. 21, p. 211909, May 2013. [Online]. Available: http://aip.scitation.org/doi/ full/10.1063/1.4808350 [29] H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, “Broadband and high-efficiency conversion from guided waves to spoof surface plasmon polaritons,” Laser Photon. Rev., vol. 8, no. 1, pp. 146–151, Jan. 2014. [Online]. Available: http://onlinelibrary.wiley.com/doi/10.1002/ lpor.201300118/abstract [30] Y. Liang, H. Yu, J. Zhao, W. Yang, and Y. Wang, “An energy efficient and low cross-talk CMOS sub-THz I/O with surface-wave modulator and interconnect,” in Proc. IEEE/ACM Int. Symp. Low Power Electron. Design (ISLPED), Jul. 2015, pp. 110–115. [31] Y. Liang, H. Yu, H. C. Zhang, C. Yang, and T. J. Cui, “Onchip sub-terahertz surface plasmon polariton transmission lines in CMOS,” Sci. Rep., vol. 5, Oct. 2015, Art. no. 14853. [Online]. Available: http://www.nature.com/srep/2015/151008/srep14853/full/ srep14853.html [32] Y. Liang et al., “On-chip sub-terahertz surface plasmon polariton transmission lines with mode converter in CMOS,” Sci. Rep., vol. 6, Jul. 2016, Art. no. 30063. [Online]. Available: http://www.nature.com/srep/2016/160721/srep30063/full/srep30063.html [33] R. Han and E. Afshari, “Filling the terahertz gap with sand: High-power terahertz radiators in silicon,” in Proc. IEEE Bipolar/BiCMOS Circuits Technol. Meeting (BCTM), Oct. 2015, pp. 172–177. [34] R. Han et al., “A 320GHz phase-locked transmitter with 3.3mW radiated power and 22.5dBm EIRP for heterodyne THz imaging systems,” in IEEE ISSCC Dig. Tech. Papers., Feb. 2015, pp. 1–3. [35] D. A. B. Miller, “Attojoule optoelectronics for low-energy information processing and communications,” J. Lightw. Technol., vol. 35, no. 3, pp. 346–396, Feb. 1, 2017. [36] D. A. B. Miller, “Optical interconnects to electronic chips,” Appl. Opt., vol. 49, no. 25, pp. F59–F70, Sep. 2010. [Online]. Available: https://www.osapublishing.org/abstract.cfm?uri=ao-49-25-F59 [37] C. Multiphysics. COMSOL Multiphysics Reference Manual, Version 5.2a. [Online]. Available: https://www.comsol.com/ [38] M. Erementchouk, S. R. Joy, and P. Mazumder, “Electrodynamics of spoof plasmons in periodically corrugated waveguides,” Proc. Roy. Soc. A, vol. 472, no. 2195, p. 20160616, Nov. 2016. [Online]. Available: http://rspa.royalsocietypublishing.org/content/472/2195/20160616 [39] A. Rusina, M. Durach, and M. I. Stockman, “Theory of spoof plasmons in real metals,” Appl. Phys. A, Solids Surf., vol. 100, no. 2, pp. 375–378, Aug. 2010. [Online]. Available: https://link.springer.com/article/10.1007/s00339-010-5866-y [40] K. Song and P. Mazumder, “Active terahertz spoof surface plasmon polariton switch comprising the perfect conductor metamaterial,” IEEE Trans. Electron Devices, vol. 56, no. 11, pp. 2792–2799, Nov. 2009. [41] J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: A comparison of latency, crosstalk and energy costs,” Opt. Express, vol. 15, no. 8, pp. 4474–4484, Apr. 2007. [Online]. Available: https:// www.osapublishing.org/oe/abstract.cfm?uri=oe-15-8-4474 [42] S.-L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightw. Technol., vol. 5, no. 1, pp. 5–15, Jan. 1987. [43] W.-P. Huang, “Coupled-mode theory for optical waveguides: An overview,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 11, no. 3, pp. 963–983, Mar. 1994. [Online]. Available: https://www.osapublishing.org/abstract.cfm?uri=josaa-11-3-963 [44] D. M. Pozar, Microwave Engineering, 4th ed. Hoboken, NJ, USA: Wiley, 2011.

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Soumitra Roy Joy received the B.Sc. and M.Sc. degrees in electrical engineering from the Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. He is currently pursuing the Ph.D. degree as a Graduate Student Research Assistant with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA. His current research interest includes subwavelength optics in terahertz spectrum.

Mikhail Erementchouk received the Ph.D. degree in physics from the City University of New York, New York, NY, USA, in 2005. He is currently a Visiting Researcher with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA. His current research interests include quantum and classical optics, optical response of semiconductors, and transport in complex media.

Hao Yu (M’06–SM’14) received the B.S. degree from Fudan University, Shanghai, China, and the Ph.D. degree from the Department of Electrical Engineering, University of California, CA, USA. He was with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is currently with the Southern University of Science and Technology, China. His primary research interests are CMOS emerging technology for data sensor, link, and accelerator. He has been the Distinguished Lecturer of the IEEE CAS since 2017. He is currently an Associate Editor and a Technical Program Committee Member for a number of journals, including Nature Scientific Reports, the IEEE T RANSACTIONS ON B IOMEDICAL C IRCUITS AND S YSTEMS , ACM Transactions on Embedded Computing System, Microelectronics Journal (Elsevier), and Integration, the VLSI Journal, and conferences, including DAC, ASSCC, DATE, and ISLPED. Pinaki Mazumder (S’84–M’87–SM’95–F’99) received the Ph.D. degree from the University of Illinois at Urbana–Champaign, Urbana, IL, USA, in 1988. He has served in industrial research and development centers including AT&TBell Laboratories, Murray Hill, NJ, USA, where he started the CONES Project called the first C modeling-based very large scale integration (VLSI) synthesis tool, and Bharat Electronics Ltd., Bengaluru, India, in 1985, where he had developed several high-speed and high-voltage analogintegrated circuits intended for consumer electronics products. He is currently a Professor with the Department of Electrical Engineering and Computer Science, University of Michigan. He has authored over 200 technical papers and 4 books on various aspects of VLSI research works. His current research interests include current problems in nanoscale CMOS VLSI design, CAD tools, and circuit designs for emerging technologies, including quantum MOS and resonant tunneling devices, semiconductor memory systems, and physical synthesis of VLSI chips. He is a fellow of the American Association for the Advancement of Science in 2008. He received the Digital Incentives for Excellence Award, the BF Goodrich National Collegiate Invention Award, and the Defense Advanced Research Projects Agency Research Excellence Award.